STRAIN
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Deformation
Strain
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Normal Strain
Shear Strain
Cartesian Strain Components
1
Deformation
θ´
B´
C´
C
θ B
A
A´
u(A)
Deformed body
Undeformed body
2
Strain
• Normal Strain
y
y
x
z
x
z
y
y
εy
1
1
x
1
εx
x
1
3
• Shear Strain
y
y
γ xy
x
π
2
x
y
εy
1
γ xy
x
1
εx
4
Example 1
The slender rod shown subjected to an increase of temperature along its axis,
which creates a normal strain in the rod of εz = 40(10-3)z1/2, where z is given in
meters. Determine (a) the displacement of the end B of the rod due to the
temperature increase, and (b) the average normal strain in the rod.
A
z
dz
200 mm
B
5
A
(a) the displacement of the end B of the rod due to the
temperature increase
z
dz
200 mm
dz ' = [40(10 −3 ) z1/ 2 ]dz
∆B = ∫
0.2 m
0
[40(10 −3 ) z1/ 2 ]dz
2
= 40(10 )( z 3 / 2 )
3
−3
B
0.2 m
0
= 0.00239 m ↓
(b) the average normal strain in the rod.
ε avg =
∆s '− ∆s 2.39 mm
=
= 0.0119 mm / mm
∆s
200 mm
6
Example 2
A force acting on the grip of the lever arm shown causes the arm to rotate
clockwise through an angle of θ = 0.002 rad. Determine the average normal
strain developed in the wire BC.
C
B
2L
L
A
7
2L
C
B B´
C
B
2L
Lsin θ
2L
L
A
Lcosθ
θ
L
A
• Solution
Since θ is small, the stretch in wire CB, L sin θ ≈ Lθ , is using Eq.2-6. The average normal
strain in the wire is then
ε avg =
CB '−CB Lθ θ 0.002
=
= =
= 0.001
CB
2L 2
2
8
Example 3
The plate is deformed into the dashed shape shown. If in this deformed shape
horizontal lines on the plate remain horizontal and do not change their length,
determine (a) the average normal strain along the side AB, and (b) the average
shear strain in the plate relative to the x and y axes.
y
3 mm
B
2 mm
250 mm
A
300 mm
C
x
9
y
(a) the average normal strain along the side AB
3 mm
AB ' = ( 250 − 2 ) 2 + ( 3) 2 = 248 . 018 mm
B
2 mm
B´
(ε AB ) avg =
250 mm
AB '− AB 248.018 − 250
=
AB
250
= −7.93(10 −3 ) mm/mm
A
(b) the average shear strain in the plate relative
to the x and y axes.
y
3 mm
B
2 mm
250 mm
γ xy =
B´
γ xy = tan −1 (
γxy
θ´
A
300 mm
C
π
2
−θ '
3
) = 0 . 0121 rad
250 − 2
x
10
Example 4
The plate shown is held in the rigid horizontal guides at its top and bottom, AD
and BC. If its right side CD is given a uniform horizontal displacement of 2 mm,
determine (a) the average normal strain along the diagonal AC, and (b) the shear
strain at E relative to the x, y axes.
x
y
D
A
150 mm
E
B
C
150 mm
2 mm
11
x
y
76 mm 76 mm
D
A
D´
A
75 mm
150 mm
θ´
E
E´
75 mm
B
C
150 mm
2 mm
B
C´
(a) the average normal strain along the diagonal AC
AC = ( 0 . 150 ) 2 + ( 0 . 150 ) 2 = 0 . 21213 m
AC ' = ( 0 . 150 ) 2 + ( 0 . 152 ) 2 = 0 . 21355 m
(ε AC ) avg =
AC '− AC 0.21355 − 0.21213
=
= 0.00669 mm/mm
AC
0.21213
12
x
y
76 mm 76 mm
D
A
D´
A
75 mm
150 mm
θ´
E
E´
75 mm
B
C
150 mm
2 mm
B
C´
(b) the shear strain at E relative to the x, y axes.
θ ' 76
tan( ) =
2
75
θ ' = 90 . 759 o =
γ xy =
π
( 90 . 759 o ) = 1 . 58404 rad
o
180
π
− 1 . 58404 = −0 . 0132 rad
2
13